Priestley duality and representations of recurrent dynamics

The paper proves that for a continuous, proper dynamical system on a compact space, the recurrent components RC(φ) are Priestley-homeomorphic and order-isomorphic to the prime spectrum ΣAtt(φ) via explicit mutually inverse maps Φ and Ψ (Lemma 6.2; Theorem 6.4), and shows the quotient topology matches the Priestley topology (Theorem 6.8). It further identifies RC(φ) as a profinite inverse limit of finite Morse representations (ΣAtt(φ) ≅ lim← ΣA and RC(φ) ≅ lim← MA; Eq. (5.3), Theorem 6.9). The model’s solution defines the same Φ, Ψ, proves they are inverse order-preserving maps, matches the basic opens j(A) \ j(A′) under Φ, and derives the profinite inverse limit through finite sublattices and restriction of prime ideals. Apart from minor differences in presentation (the model appeals to standard Conley facts and uses a direct clopen-preimage argument; the paper routes continuity through Morse sets), the arguments coincide in substance and hypotheses.